A uniqueness result for the decomposition of vector fields in Rd

Given a vector field \u3c1(1,b) 08Lloc1(R+ 7Rd,Rd+1) such that divt,x(\u3c1(1,b)) is a measure, we consider the problem of uniqueness of the representation \u3b7 of \u3c1(1 , b) Ld+1 as a superposition of characteristics \u3b3:(t\u3b3-,t\u3b3+)\u2192Rd, \u3b3\u2d9 (t) = b(t, \u3b3(t)). We give conditions in terms of a local structure of the representation \u3b7 on suitable sets in order to prove that there is a partition of Rd+1 into disjoint trajectories \u2118a, a 08 A, such that the PDE divt,x(u\u3c1(1,b)) 08M(Rd+1),u 08L 1e(R+ 7Rd),can be disintegrated into a family of ODEs along \u2118a with measure r.h.s. The decomposition \u2118a is essentially unique. We finally show that b 08Lt1(BVx)loc satisfies this local structural assumption and this yields, in particular, the renormalization property for nearly incompressible BV vector fields

FORWARD UNTANGLING AND APPLICATIONS TO THE UNIQUENESS PROBLEM FOR THE CONTINUITY EQUATION

We introduce the notion of forward untangled Lagrangian representation of a measure-divergence vector-measure rho(1, b), where rho is an element of M+(Rd+1) and b : Rd+1 -> R-d is a rho-integrable vector field with div(t,x)(rho(1, b)) = mu is an element of M(R x R-d): forward untangling formalizes the notion of forward uniqueness in the language of Lagrangian representations. We identify local conditions for a Lagrangian representation to be forward untangled, and we show how to derive global forward untangling from such local assumptions. We then show how to reduce the PDE div(t,x)(rho(1, b)) = mu on a partition of R+ x R-d obtained concatenating the curves seen by the Lagrangian representation. As an application, we recover known well posedeness results for the flow of monotone vector fields and for the associated continuity equation

Sistematización de la Feria Provincial de Semillas Nativas y Criollas, “Sembrando Esperanza” : Un Aporte al Intercambio de Saberes

Desde 2007 se realiza durante el mes de mayo en el Parque Pereyra Iraola, Buenos Aires (Argentina), la Feria de Semillas Nativas y Criollas, espacio de encuentro e intercambio que realza la importancia de producir y conservar semillas, haciendo visible el papel de las organizaciones de agricultores familiares en la conservación de la biodiversidad. En este marco creamos un grupo de trabajo con la finalidad de registrar la diversidad de productos y materiales que circularon y dar a conocer la variedad de semillas de las organizaciones. En 2008 el relevamiento se realizó en los puestos de las organizaciones y productores presentes mediante entrevistas y toma de muestras. Observamos 392 muestras de semillas y partes reproductivas en puestos de 37 organizaciones. En la Feria 2009 presentamos el Libro de las Semillas que revela lo ocurrido en los talleres, paneles e intercambio en los stands y las semillas presentes en la Feria 2008, y realizamos un nuevo relevamiento en proceso de análisis. El objetivo de este trabajo es compartir esta experiencia.Facultad de Ciencias Agrarias y Forestale

Analysis of an integral equation arising from a variational problem

Rapporto Interno, Dipartimento di Matematica, Politecnico di Torin

Analysis of an integral equation arising from a variational problem

Rapporto Interno, Dipartimento di Matematica, Politecnico di Torin

Superposition principle for the continuity equation in a bounded domain

We consider an initial-boundary value problem for the continuity equation in a class of non-negative measure-valued solutions. We prove that any solution in the considered class can be represented as a superposition of elementary solutions, associated with the solutions of the corresponding ordinary differential equation. © Published under licence by IOP Publishing Ltd

Regularity estimates for the flow of BV autonomous divergence-free vector fields in R^2

We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in time. As a consequence we deduce that both geometric and analytical mixing have a lower bound of order (Formula presented.) as (Formula presented.)